Untitled

A partial cyclic prefix based method for frequency offset estimation with
robustness against symbol timing offset
Ning Han Soung-Hwan Shon Jae-Moung Kim
neil_han@hotmail.com, kittisn@naver.com, jaekim@inha.ac.kr
Graduate school of Information Technology & Telecommunications Inha University
Abstract
In this paper, we propose an improved method based on J.J. van Beek algorithm to estimate the frequency offset
in OFDM systems. The improved estimation method uses part of the cyclic prefix, therefore, by carefully choosing
the length of CP, it has good robustness against symbol timing offset, at the same time, the estimation accuracy can be
guaranteed. Simulation results show that the mean-squared error (MSE) can be significantly improved at high Eb/N0
level with the proposed method compared with the conventional one.
Ⅰ. Introduction
Orthogonal frequency division multiplexing (OFDM) is an
effective transmission technology to combat multipath fading,
by circular copying the last part of the symbol as the prefix, the
inter-symbol interference (ISI), as well as the inter-carrier
interference (ICI) can be mitigated.
OFDM systems are very sensitive to carrier frequency offset
and even a small frequency offset can greatly degrade the
performance of the OFDM receiver [1]. There are two kinds of
frequency offset estimation algorithms, data-aided and
non-data-aided. Many data-aided estimators rely on the
periodical transmission of known data blocks to correct the
carrier frequency offset [1-4]. These estimators ensure fast and
reliable synchronization, but they reduce system resource
utilization. Blind estimation and cyclic prefix based estimation
algorithm are non-data-aided, but blind estimation increases the
system complexity greatly. J.J. van Beek proposed a method to
estimate fractional frequency offset in time domain. It uses the
redundant information contained within the cyclic prefix [5].
In this paper, an improved fractional frequency offset
estimation method is proposed which uses just part of the CP
length. Compared to J. J. van Beek algorithm, the proposed
scheme has almost the same estimation accuracy, but higher
robustness against symbol timing offset.
The remainder of this paper is organized as follow. SectionⅡ
briefly introduces the OFDM fundamentals. SectionⅢ presents
the improved carrier frequency offset estimation method. Some
simulation results follows in sectionⅣ. At last, some conclusions
are drawn in sectionⅤ.
Ⅱ. OFDM System Model
OFDM input signals are complex numbers of some signal
constellation (e.g., PSK, QAM). Let X(k) denotes complex signal
modulated on to the k-th subcarrier. The output signal samples, x(n)
can be found by an Inverse Discrete Fourier Transform (IDFT) of
X(k), as given by:
x
1 N
N
∑ −1
k = 0
( n ) = X ( k )
e
π n=0, 1, 2,…, N-1 (1)
j 2 kn / N
After that, the last L samples of x(n) are copied and put as a
preamble (Cyclic Prefix) to form one OFDM Symbol. The length
L of cyclic prefix should be longer than the channel impulse
response, in order to avoid ISI.
Then, passing x(n) through the channel, we obtain the complex
received signals as given by:
y
N
= ∑ −1
j 2π ( k + ε ) n / N
( n)
X ( k ) H ( k ) e + w(
n)
k = 0
n=-L, - (L-1), … , 0, 1, 2, … , N-1 (2)
where H(k) is a complex transfer function of the channel at the
frequency of the k-th subcarrier, ε is the frequency offset
normalized to the subcarrier spacing, and w(n) denotes the
samples of the complex envelope of AWGN. The corresponding
baseband signal is given by:
y
j2π
nε
/ N
( n)
= x(
n)
e + w(
n)
(3)

The received samples, after the cyclic prefix is removed, can be
turned into the estimate, Y(k), of the transmitted signals, X(k), by
a Discrete Fourier Transform (DFT) , given by:
Y
1
N
N 1
π
− j 2 nk / N
( k ) y(
n ) e + W ( k )
= ∑ −
n = 0
k=0, 1, 2, … , N-1 (4)
a). J.J. van Beek algorithm:
Ⅲ. Frequency offset estimation
The conventional J.J. van Beek algorithm uses the cyclic
property of the CP and the correlation length is L which is the
length of CP. The performance can be guaranteed if no symbol
timing errors occur. The symbol structure is shown in Fig.1:
Fig.1. OFDM symbol structure for J.J. van Beek algorithm, no
symbol timing offset
Therefore, the frequency offset can be estimated as given by:
∧ 1
ε = − ∠ γ ( m )
(5)
2π
Where the parameter m denotes the symbol timing offset, γ ( m)
is
the correlation function, given by:
γ
∑ − + m L 1
n = m
∗
( m ) = y(
n ) y ( n + N )
∧
We can find that the estimated offset depends on the symbol
timing offset number m.
When symbol timing errors occur, the colored part (as shown in
Fig.1) named as moving sum window for correlation moves
backward (as shown in Fig.2).
Fig.2. OFDM symbol structure for J.J. van Beek algorithm,
ε
symbol timing offset number is m.
The first part maintains the cyclic property, but rests of the moving
window losses this property. This is the reason why the
(6)
performance is poor when symbol timing errors occur.
Simulations in AWGN and Rayleigh fading channel are made.
(as shown in Fig.3)
Simulation result shows, that if the previous timing estimation
procedure is accurate or there is no symbol timing offset, the
performance of this algorithm is good (Fig.3). But the
performance decreases rapidly even small number of symbol
timing errors occur. Fig.4 shows the performance degradation in
both AWGN and Rayleigh fading channel as a function of symbol
timing offset. Even the Eb/N0 value is large, the performance
degrades rapidly when the symbol timing errors increases to about
40 samples.
MSE
MSE
1.00E+00
1.00E-01
1.00E-02
1.00E-03
1.00E-04
1.00E-05
1.00E-06
1.00E-07
1.00E-08
1.00E-09
AWGN Channel
Rayleigh Fading Channel
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51
Eb/N0
Fig.3. MSE versus En/N0 for the J.J van Beek algorithm in
1.0E+00
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06
1.0E-07
AWGN and Rayleigh fading channel.
AWGN 0dB
AWGN 25dB
Rayleigh Fading 25dB
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96
Symbol Timing Offset
Fig.4. MSE versus Symbol timing offset for the J.J van Beek
algorithm in AWGN and Rayleigh fading channel.
b). Proposed method
We propose an improved method to estimate the frequency
offset by using part of the CP length. When symbol timing offset
occurs, the OFDM symbol structure of the proposed method is

shown below.
Fig.5. OFDM symbol structure for the proposed method,
symbol timing offset number is m.
The corresponding correlation function is given by:
∑ − + m β L 1
n = m
∗
( m ) = y(
n ) y ( n + N )
γ (7)
Whereβis the CP used rate. It could be 1/2, 1/4 and even 1/8.
We can find that as the moving sum window length decreases, the
cyclic property can be maintained. When the moving sum window
boundary does not extend beyond the CP length, good
performance can be guaranteed.
As the CP used rate β decreases, the moving sum window
length decreases ( βL in (7)), therefore the estimation accuracy
gets worse. The results can be proved by the simulation results in
the following section.
Ⅳ. Computer simulations
An outdoor dispersive, fading environment is selected and
modeled as a channel consisting of 6 independent Rayleigh fading
paths. The maximum vehicle speed is 100 km/h. The total
subcarrier number is 2048. A cyclic prefix with length of 128 is
used. The proposed method can be realized by the following
structure (Fig.6).
Fig.6. Implementation structure of the proposed method.
We first assumed that frequency offset is 0.38, and symbol timing
offset is 50 samples, comparing the mean-squared error (MSE)
performance of the improved method with the conventional
algorithm in AWGN channel with different Eb/N0 value.The
parameterβis set to 1 for the J.J van Beek algorithm, and 1/2, 1/4,
1/8 for the improved method respectively. Fig.7 shows the
performance.
MSE
1.00E+00
1.00E-01
1.00E-02
1.00E-03
1.00E-04
1.00E-05
1.00E-06
1.00E-07
1.00E-08
1.00E-09
0
4
8
12
16
20
24
28
Eb/N0
32
36
40
44
β= 1
β= 1/2
β= 1/4
β= 1/8
Fig.7. MSE versus Eb/N0 performance for the comparison of the
improved method and the J.J Van Beek algorithm in AWGN
channel with 50 samples symbol timing offset
In Fig.7, when symbol timing error occurs, increasing the
Eb/N0 value can not improve the performance for the
conventional J.J van Beek algorithm. But, it has a different result
for the proposed method. As the Eb/N0 value increasing the
performance increases linearly.
The performance of the proposed method in fading channel is
shown below in Fig.8. As the fading path number increasing, the
distortion caused by the delayed paths increase, therefore, the
performance of 2-path fading is better than that of 6-path.
MSE
1.00E+00
1.00E-01
1.00E-02
1.00E-03
1.00E-04
1.00E-05
1.00E-06
1.00E-07
1.00E-08
1.00E-09
1.00E-10
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51
Eb/N0
48
β=1/4 6-path
β=1/4 2-path
β=1/2 6-path
β=1/2 2-path
Fig.8. MSE versus Eb/N0 performance for the improved method
in 2-path and 6-path Rayleigh fading channel with 50 samples
symbol timing offset and differentβvalue
Then we assumed that the frequency offset is 0.38, comparing
the robustness of the proposed method with the conventional
algorithm in both AWGN and Rayleigh fading channel.
Fig.9, .10, .11 show the simulation results.

MSE
1.0E+00
1.0E-01
1.0E-02
1.0E-03
1.0E-04
β= 1
β= 1/2
β= 1/4
β= 1/8
0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144
Symbol Timing Offset
Fig.9. MSE versus Symbol Timing Offset performance in AWGN
MSE
1.0E+00
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06
channel with Eb/N0 equals to 0dB.
β= 1
β= 1/2
β= 1/4
β= 1/8
0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144
Symbol Timing Offset
Fig.10. MSE versus Symbol Timing Offset performance in
AWGN channel with Eb/N0 equals to 20dB.
In Fig.9, the proposed method performs not stably. As the
Eb/N0 value increases, as shown in Fig.10, the proposed method
has a better performance, and even in Rayleigh fading channel the
proposed method has a good robustness against symbol timing
offset, as shown in Fig.11. We can also find that as the CP used
rate β decreases, the proposed method have better robustness
against symbol timing offset, but the accuracy decreased only a
little.
All the simulation results prove our analysis in the previous
section. The proposed method does have the robustness against,
and the estimation accuracy can be guaranteed at the same time.
Ⅴ. Conclusions
In this paper, an improved method of frequency offset
estimation is proposed, based on the conventional J.J. van Beek
algorithm, which utilizes the correlation property of the cyclic
prefix in the time domain.
MSE
1.0E+00
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06
1.0E-07
1.0E-08
β= 1
β= 1/2
β= 1/4
β= 1/8
0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144
Symbol Timing Offset
Fig.11 MSE versus Symbol Timing Offset performance in
Rayleigh fading channel with Eb/N0 equals to 25dB.
The proposed method utilizes part of the CP length; therefore, it
is robust against the symbol timing offset. Simulation results
shows the robustness depends on the CP used rate β. Also, the
accuracy decreases a little bit as β decreases. We think this method
can be implemented in the situation that small number of symbol
timing errors occurs, or after the symbol timing offset estimation.
Future work should focus on the forward symbol timing
estimation researching and a data-aided method study to guarantee
higher estimation accuracy.
This research was supported by University IT Research Center
Project (INHA UWB-ITRC), Korea
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